# The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of the presheaf $U\mapsto H^i_{et}(U,{\mathbb{Z}}/l {\mathbb{Z}} (r))$ ($U$ runs through open subvarieties of $X$) has a flabby resolution given by the corresponding Cousin complex; this is called the Gersten resolution. This results seems to imply its Nisnevich analogue easily.

Now, let $i:Z\to X$ be a closed embedding. Did anybody study what happens with the Gersten resolution when we apply (the Nisnevich-sheaf) $i^*$ to it? Certainly, the case of a smooth $Z$ should be easier; yet I also wonder what can be said if $Z$ is a normal crossing divisor. Moreover, are there any 'base-change-like' results for this situation (note that we have a cartesian square of sites: $X_{et}$, $X_{nis}$, $Z_{et}$, $Z_{nis}$, and there are also sites over the Zariski points of $X$ in this picture)? One can apply rigidity (for homotopy invariant sheaves with transfers) in order to study the cohomology of the Cousin complex; yet I wonder whether anybody did this already, and what else could be said.

-